Compare the graphs of the logarithmic functions f left parenthesis x right parenthesis equals log subscript 7 baseline x and g left parenthesis x right parenthesis equals log subscript 4 baseline x. For what values of x is f = g, f greater than g, and f less than g? Explain how you know.

Question

Compare the graphs of the logarithmic functions f left parenthesis x right parenthesis equals log subscript 7 baseline x and g left parenthesis x right parenthesis equals log subscript 4 baseline x. For what values of x is f = g, f greater than g, and f less than g? Explain how you know.
(4 points)

GPT-4o mini · Answer

To compare the logarithmic functions $ f(x) = \log_7(x) $ and $ g(x) = \log_4(x) $, we need to analyze their graphs and how they behave over various intervals of $ x $. ### Understanding the Functions 1. **Base of the Logarithm**: - The function $ f(x) $ is a logarithm with base 7, and $ g(x) $ is a logarithm with base 4. Since 7 is larger than 4, we can expect that for any positive $ x $, $ \log_7(x) $ will grow more slowly than $ \log_4(x) $. 2. **Intercepts**: - Both functions have an intercept at (1, 0) since $ \log_b(1) = 0 $ for any base $ b $. ### Identifying Points of Intersection To find the values of $ x $ where $ f(x) = g(x) $: $$ \log_7(x) = \log_4(x) $$ Using the change of base formula, we can rewrite this as: $$ \frac{\log(x)}{\log(7)} = \frac{\log(x)}{\log(4)} $$ Cross multiplying gives: $$ \log(x) \cdot \log(4) = \log(x) \cdot \log(7) $$ If $ \log(x) \neq 0 $ (which is true for $ x \neq 1 $): $$ \log(4) = \log(7) \quad \text{(which is false)} $$ So, the only point where $ f(x) = g(x) $ is at $ x = 1 $. ### Comparing the Functions Now, we can analyze when $ f(x) $ is greater than or less than $ g(x) $: 1. **For $ x < 1 $**: - Both functions are negative because logarithms of numbers less than 1 (but greater than 0) are negative. However, since $ \log(4) > \log(7) $, $ g(x) < f(x) $ in this interval. Thus, for $ 0 < x < 1 $: $$ f(x) > g(x) $$ 2. **For $ x = 1 $**: - As established before, $ f(1) = g(1) = 0 $: $$ f(x) = g(x) $$ 3. **For $ x > 1 $**: - Both functions are positive. The growth rate of $ g(x) = \log_4(x) $ will be greater than that of $ f(x) = \log_7(x) $ due to the smaller base of the logarithm. Therefore, for $ x > 1 $: $$ f(x) < g(x) $$ ### Summary - **Values of $ x $ where $ f = g $**: $ x = 1 $ - **Values of $ x $ where $ f > g $**: $ 0 < x < 1 $ - **Values of $ x $ where $ f < g $**: $ x > 1 $ ### Conclusion The graphs of the functions $ f(x) = \log_7(x) $ and $ g(x) = \log_4(x) $ intersect at $ x = 1 $. For values less than 1, $ f $ is greater than $ g $, and for values greater than 1, $ f $ is less than $ g $. This behavior is dictated by the bases of the logarithms: $ \log_4(x) $ outpaces $ \log_7(x) $ as $ x $ increases.